Average Error: 0.2 → 0.2
Time: 6.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)
double f(double B, double x) {
        double r12852 = x;
        double r12853 = 1.0;
        double r12854 = B;
        double r12855 = tan(r12854);
        double r12856 = r12853 / r12855;
        double r12857 = r12852 * r12856;
        double r12858 = -r12857;
        double r12859 = sin(r12854);
        double r12860 = r12853 / r12859;
        double r12861 = r12858 + r12860;
        return r12861;
}
↓
double f(double B, double x) {
        double r12862 = x;
        double r12863 = 1.0;
        double r12864 = r12862 * r12863;
        double r12865 = B;
        double r12866 = sin(r12865);
        double r12867 = r12864 / r12866;
        double r12868 = -r12867;
        double r12869 = cos(r12865);
        double r12870 = r12863 / r12866;
        double r12871 = fma(r12868, r12869, r12870);
        return r12871;
}

Error

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
  2. Using strategy rm

  3. Applied associate-*r/0.2

    \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B}\]
  4. Using strategy rm

  5. Applied tan-quot0.2

    \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\frac{\sin B}{\cos B}}}\right) + \frac{1}{\sin B}\]

  6. Applied associate-/r/0.2

    \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\sin B} \cdot \cos B}\right) + \frac{1}{\sin B}\]

  7. Applied distribute-lft-neg-in0.2

    \[\leadsto \color{blue}{\left(-\frac{x \cdot 1}{\sin B}\right) \cdot \cos B} + \frac{1}{\sin B}\]

  8. Applied fma-def0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)}\]

  9. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))